Radar rainfall estimation technique when attenuation is negligible

ABSTRACT

The invention provides a method of using a bipolar radar to estimate a precipitation rate for rain that generates negligible attenuation, the method being constituted by the following steps:  
     using said bipolar radar to measure the differential phase Φ DP  and the apparent reflectivity Z e  on at least one of horizontal polarization H and vertical polarization V over a given interval [r 0 , r 1 ] of path radii relative to said radar;  
     making an estimate of the value N 0 * representative of the dimensional distribution of rain drops on the basis of the differential phase difference in the range r 0  to r 1  and on the basis of an integral of a function of the apparent reflectivity Z e  along the interval [r 0 , r 1 ]; and  
     deducing a value for the precipitation rate at a point from N 0 * and the apparent reflectivity at said point.

[0001] The present invention relates to meteorological techniques forestimating a precipitation rate by means of radar.

[0002] When estimating the rate of rainfall from radar measurements, oneis generally confronted with the problem of attenuation of the radarwave, and with the problem of the natural variability of rain.

[0003] The concept of coherent radar with polarization diversityassociated with a so-called ZPHI algorithm is described in WO 99/38028as a solution for those two obstacles under operational conditions.

[0004] ZPHI is a profiler algorithm which basically uses as input ameasured reflectivity profile Z_(a) and a constraint given by adifferential phase difference Φ_(DP) between two points r₁ and r₂ on aline of sight.

[0005] On the basis of those measurements, the specific attenuation Aand a parameter known as N₀* are determined, which parameter quantifiesthe distribution of raindrop sizes.

[0006] The rainfall rate R to be estimated is obtained as a function ofthose two parameters, for the following two reasons:

[0007] the specific attenuation A is not subject to attenuation effects.Using it to estimate R thus enables the attenuation problem to beovercome; and

[0008] the parameter N₀* suffices to describe the natural variability ofrain.

[0009] The ZPHI algorithm is equally applicable in the X and C bands,which frequency bands are sensitive to attenuation.

[0010] The present specification relates to a method extended from theZPHI method when attenuation is small.

[0011] In this particular circumstance, it is possible to specify aformalism that is based on base elements, but that is better adapted andmuch more direct.

[0012] The ZPHI algorithm can also be applied in the S band, whereattenuation is small, or indeed negligible. The advantage of so doinglies in reproducing the natural variability of rain “in real time”. Useof the parameter A is then no longer strictly necessary.

[0013] An object of the invention is thus to propose a method that issimple, reliable, and effective for evaluating a precipitation rate whenattenuation is weak or negligible.

[0014] For this purpose, the invention provides a method of using abipolar radar to estimate a precipitation rate for rain that generatesnegligible attenuation, the method being constituted by the followingsteps:

[0015] using said bipolar radar to measure the differential phase Φ_(DP)and the apparent reflectivity Z_(e) on at least one of horizontalpolarization H and vertical polarization V over a given interval [r₀,r₁] of path radii relative to said radar;

[0016] making an estimate of the value N₀* representative of thedimensional distribution of rain drops on the basis of the differentialphase difference in the range r₀ to r₁ and on the basis of an integralof a function of the apparent reflectivity Z_(e) along the interval [r₀,r₁]; and

[0017] deducing a value for the precipitation rate at a point from N₀*and the apparent reflectivity at said point.

[0018] The invention also provides an apparatus for estimating aprecipitation rate, the apparatus comprising a bipolar radar andprocessor means, said radar including means for measuring differentialphase and reflectivity on at least one of horizontal polarization H andvertical polarization V, the apparatus being characterized in that theprocessor means implement the various processing steps of the methodaccording to the above paragraph.

[0019] Other characteristics, objects, and advantages of the inventionappear on reading the following detailed description.

[0020] As in the ZPHI method, the method described below requires acoherent polarization diversity radar. The input data are thereflectivity profile Z_(H) (or Z_(V)), together with a measurement ofthe differential phase between the H channel and the V channel.

[0021] The purpose here is to determine the parameter N₀* (and thus therainfall rate) when attenuation is negligible, for example for rainmeasured in the S band, for rain that is sufficiently light and measuredin the C and X bands, or indeed for ice.

[0022] As in the ZPHI method, the present method makes use of two typesof data: reflectivity Z_(H) (or Z_(V), or more generally Z_(e)) anddifferential phase Φ_(DP).

[0023] Providing attenuation is negligible, the measured reflectivitycan be used directly to estimate rainfall rate R on the basis of anestimate for the parameter N₀* which specifically constitutes thesubject matter of the present method.

[0024] The method relies on the “universal” relationship associatingequivalent reflectivity Z_(e) (mm⁶m⁻³) and differential phase rateK_(DP) (° km⁻¹): $\begin{matrix}{\frac{K_{DP}}{N_{0}^{*}} = {a\left\lbrack \frac{Z_{e}}{N_{0}^{*}} \right\rbrack}^{b}} & (1)\end{matrix}$

[0025] Where a and b are coefficients specified by the diffusion model,and depend on the type of precipitation (rain or ice crystal type) andare also functions of temperature to a small extent. The type ofprecipitation can be determined by a classification method of the typedescribed by Straka, Zrnié, and Ryzhkov (2000).

[0026] More generally, the diffusion model defines the relationship:$\begin{matrix}{\frac{K_{DP}}{N_{O}^{*}} = {F\left\lbrack \frac{Z_{e}}{N_{0}^{*}} \right\rbrack}} & (2)\end{matrix}$

[0027] By integrating (1) or (2) between the two bounds r₁ and r₂ of theintegration segment (r₁<r₂) the following is obtained:

Φ_(DP)(r ₂)−Φ_(DP)(r ₁)=[N ₀*]^(1−b)∫_(r1) ^(r2) Z _(e) ^(b) ds  (3)

[0028] I.e.: $\begin{matrix}{N_{0}^{*} = \left\lbrack \frac{{\Phi_{DP}\left( r_{2} \right)} - {\Phi_{DP}\left( r_{1} \right)}}{a{\int_{r1}^{r2}{Z_{e}^{b}\quad {s}}}} \right\rbrack^{\frac{1}{1 - b}}} & (4)\end{matrix}$

[0029] All that is then required in order to obtain the estimate for Ris to feed this estimate for N₀* into the “universal” relationshipassociating R with Z_(e):

R=c└N ₀*┘^(1−d) Z _(e) ^(d)  (5)

[0030] In the general case where the relationship K_(DP)−Z_(e) iswritten in the form (2), (3) becomes: $\begin{matrix}{{{\Phi_{DP}\left( r_{2} \right)} - {\Phi_{DP}\left( r_{1} \right)}} = {N_{0}^{*}{\int_{r1}^{r2}{{F\left( \frac{Z_{e}}{N_{0}^{*}} \right)}\quad {s}}}}} & (6)\end{matrix}$

[0031] An analytic solution is not then available for N₀*, but anumerical solution can be found by an iterative numerical method usingsolution (4) as the first guess for N₀*.

[0032] For ice, which hardly attenuates the radar wave at all regardlessof the frequency used (X, C, and S bands), there are no fixed limits onrange of application.

[0033] For rain, application of the method is constrained by therequirement for attenuation to be negligible between the points r₁ andr₂ (r₂>r₁). More specifically, application conditions leading to anerror of 3 decibels (dB) on N₀* are as follows (for T=10° C., and for agamma type relationship having a form parameter equal to 2 governingdroplet size distribution):

[0034] for X band: [Φ_(DP)(r₂)−Φ_(DP)(r₁)]=≦4°

[0035] for C band: [Φ_(DP)(r₂)−Φ_(DP)(r₁)]=≦10°

[0036] for S band: [Φ_(DP)(r₂)−Φ_(DP)(r₁)]=≦64°

[0037] The main fields of application of the invention are as follows:

[0038] the same fields of application as for the ZPHI algorithm, i.e.estimating rain in catchment areas for flood warning and for waterresource management. This application is valid for all types of rainusing S band, or to rain that is sufficiently light in C band and Xband; and

[0039] estimating precipitation in the form of ice.

[0040] The invention also presents numerous other applications inmeteorology.

[0041] [1] Differential propagation phase shift and rainfall rateestimation, M. Sachidananda and D. S. Zrnié, Radio Science, 21-2, pp.235-247 (1986).

[0042] [2] Polarimetric method for ice water content determination, A.V. Ryshkov, D. S. Zrnié, and B. A. Gordon, J. Appl. Meteor., pp. 125-134(1998).

[0043] [3] The rain profiling algorithm applied to polarimetric weatherradar, J. E. Testud, E. Le Bouar, E. Obligis, M. Ali-Mehenni, and J.Atmos. Oceanic Technol., 17, pp. 332-356 (2000).

[0044] [4] Bulk hydrometeor classification and quantification usingpolarimetric radar data: synthesis of relations, J. M. Straka, D. S.Zrnié, and A. V. Ryzhkov, J. Appl. Meteor., 39, pp. 1341-1372 (2000).

1. A method of using a bipolar radar to estimate a precipitation ratefor rain that generates negligible attenuation, the method beingconstituted by the following steps: using said bipolar radar to measurethe differential phase Φ_(DP) and the apparent reflectivity Z_(e) on atleast one of horizontal polarization H and vertical polarization V overa given interval [r₀, r₁] of path radii relative to said radar; makingan estimate of the value N₀* representative of the dimensionaldistribution of rain drops on the basis of the differential phasedifference in the range r₀ to r₁ and on the basis of an integral of afunction of the apparent reflectivity Z_(e) along the interval [r₀, r₁];and deducing a value for the precipitation rate at a point from N₀* andthe apparent reflectivity at said point.
 2. A method according to claim1, characterized in that N₀* is deduced directly from the differentialphase difference between r₀ and r₁ and from the integral of the apparentreflectivity Z_(e) raised to a selected exponent.
 3. A method accordingto claim 1, characterized in that the selected exponent is an exponent bsatisfying:$\frac{K_{DP}}{N_{O}^{*}} = {a\left\lbrack \frac{Z_{e}}{N_{0}^{*}} \right\rbrack}^{b}$

where K_(DP) is the rate of variation in the differential phase alongthe radius, a and b being specified by the type of precipitation underconsideration and by temperature.
 4. A method according to claim 3,characterized in that the type of precipitation is a corresponding typeselected from rain type and ice crystal type.
 5. A method according toclaim 1, characterized in that the apparent reflectivity function whichis integrated depends on N₀*, and in that the method includescalculating this function in a plurality of iterations using a value forN₀* as determined at the preceding iteration.
 6. A method according toclaim 5, characterized in that N₀* is calculated from the followingrelationship:${{\Phi_{DP}\left( r_{2} \right)} - {\Phi_{DP}\left( r_{1} \right)}} = {N_{0}^{*}{\int_{r1}^{r2}{{F\left( \frac{Z_{e}}{N_{0}^{*}} \right)}\quad {s}}}}$

where Φ_(DP) is the differential phase and where F is a functionsatisfying the following relationship:$\frac{K_{DP}}{N_{0}^{*}} = {F\left( \frac{Z_{e}}{N_{0}^{*}} \right)}$

where K_(DP) is the rate of variation in differential phase along theradius.
 7. A method according to claim 5, characterized in that, for thefirst iteration, the value used for N₀* is given by:$N_{O}^{*} = \left\lbrack \frac{{\Phi_{DP}\left( r_{2} \right)} - {\Phi_{DP}\left( r_{1} \right)}}{a{\int_{r1}^{r2}{Z_{e}^{b}\quad {s}}}} \right\rbrack^{\frac{1}{1 - b}}$


8. A method according to claim 1, characterized in that the rainfallrate R is determined from N₀* and from the apparent reflectivity Z_(e)by the relationship: R=c└N ₀*┘^(1−d) Z _(e) ^(d)
 9. Apparatus forestimating a precipitation rate, the apparatus comprising a bipolarradar and processor means, said radar including means for measuringdifferential phase and reflectivity on at least one of horizontalpolarization H and vertical polarization V, the apparatus beingcharacterized in that the processor means implement the variousprocessing steps of the method according to claim 1.